Isomorphisms of BV(\sigma) and AC(\sigma) Spaces.

Abstract

Abstract The Banach algebras AC(a) and BV (a) were introduced by Ashton and Doust in 2005 in order to extend the theory of well-bounded operators. Here is a nonempty compact subset of the plane and the elements of these algebras are called absolutely continuous functions and functions of bounded variation respectively. The aim of this thesis is to investigate the extent to which analogues of the classical 1939 theorem of Gelfand and Kolmogoroff for C(K) spaces might hold in the context of these functions. That is, we study the relationship between the topological structure of the domain set and the structure of the Banach algebra AC(a).

In 201S Doust and Leinert had shown that if AC(o_ 1) is isomorphic to AC (o_ 2) then o_1 must be homeomorphic to o_ 2, providing one direction of a Gelfand-Xolmogoroff type theorem. They also showed that although the converse implication fails in general, if one restricts the class of sets considered, then positive theorems are possible. In particular they showed that if o_ 1 and o_ 2 are polygonal, then AC (o_ 1) is isomorphic to AC (o_ 2) if and only if o_1 and o_ 2 are homeomorphic.

In this thesis we consider the situation for three natural classes of compact sets: those that are the spectra of compact operators. those which are the union of a finite number of line segments, and a more general class of polygonally inscribed curve. Full analogues of the Gelfand-Kolmogoroff theorem are proven for the latter two classes. Considering the first class, it is shown that there are infinitely many homeomorphic sets in that tlass with mutually non-isomorphic AC (o).

A study of isomorphisms of BV (a) spaces is also initiated. The main result is that if AC(o 1) is isomorphic to AC(o_2) then necessarily BV (o_1) is isomorphic to BV (o_2). An example is given to show that the converse of this result is not true.